Evidence against field decay proportional to accreted mass in neutron stars

8 Pages·0.26 MB·English

by  Ralph A. M. J. Wijers

Mon.Not.R.Astron.Soc.000,000–000 (0000) Printed1February2008 (MNLATEXstylefilev1.4)
Evidence against field decay proportional to accreted mass
in neutron stars
Ralph A.M.J. Wijers
Institute of Astronomy, Madingley Road, Cambridge CB3 0HA, UK
Email: [email protected]
submittedtoMNRAS,24-7-96,revised27-11-96,accepted 02-01-97
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9
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1 ABSTRACT
Aspecificclassofpulsarrecyclingmodel,inwhichmagnetic-fielddecreaseisafunction
n
only of the amount of mass accreted onto the neutron star, is examined in detail. It
a
J is shown that no model in this class is consistent with all available data on X-ray
binaries and recycled pulsars. Only if all constraints are stretched to their limit and
8
a few objects (PSRB1831−00 and 4U1626−67) are assumed to have formed in a

2 non-standard manner is there still an acceptable model of this kind left. Improved
v measurements of the parameters of a few of the oldest known radio pulsars will soon
5 test andprobably rule out that one as well. Evidence for the originofPSRB1831−00
7 via accretion-induced collapse of a white dwarf is called into question as a result.
0
8 Key words: pulsars: recycled – neutron stars: structure – neutron stars: magnetic
0 fields – X-ray binaries
6
9
/
h
p 1 INTRODUCTION 2 THE MODEL
-
o The growing body of evidence against rapid spontaneous 2.1 Neutron star structure
r
st magnetic-field decay in neutron stars (Kulkarni 1986; Ver- Theamount of mass accreted onto recycled pulsars can ap-
bunt, Wijers, & Burm 1990; Bhattacharya et al. 1992) has
a proach M⊙, so we should follow the concomitant change of
: led to the hypothesis that pulsar magnetic fields are low- radiusandmomentofinertia.Tocomputethegravitational
v
ered by recycling in binaries (see Bhattacharya & Van den
i mass,Mg,asafunctionofbaryonmass,Mb,Iusethebind-
X Heuvel1991 for a review). The mechanism for this remains
ing energy correction given byLattimer & Yahil(1989):
unclear, with suggestions including decay of crustal fields
r
a d(Buiesntoovhaetaytii-nKgo(gBanlon&diKno&mFbreeregse11997856;)Roormbaunriial19o9f0t,he19fi9e3ld) Mb =Mg+0.084M⊙(cid:18)MM⊙g(cid:19)2 (1)
anddecayof corefieldsduetofluxtubeexpulsionfrom the
superfluid interior (Srinivasan et al. 1990). AsMbincreasesbyaccretion,Mgincreasesmoreslowly.The
mass dependenceof theradius is taken to be
A heuristic model first suggested by Taam and Van
den Heuvel (1986) and explored further by Shibazaki et al. −1/3
(1989) is that the ratio of initial to final magnetic field is R6 = R = Mg (2)
10km (cid:18)1.4M⊙(cid:19)
simplyproportionaltotheamountofmassaccreted.Thede-
tailed model by Romani (1993) supports this in the regime and assuming the radius of gyration is a constant fraction
ofhighaccretionrates.VandenHeuvelandBitzaraki(1995) of R the moment of inertia will be
have shown that it is consistent with the magnetic fields of +1/3
mInotshtirsecpyacpleerdtphuelsmarosdwelit(hanhdeliaumgenwehriatleisdedwavrefrcsioomnpoafniito)nsis. I45 = 1045Igcm2 =(cid:18)1.M4Mg⊙(cid:19) . (3)
tested for agreement with a wider body of data including Here the normalisation is chosen to get the ‘standard’ val-
many typesof neutron star binary,and it fails the test. ues I45 = 1 and R6 = 1 for a 1.4M⊙ neutron star. Test
The details of the model used to evolve pulsar periods calculations show that modest deviations from the above
and fields is discussed in Section 2. The constraints derived do not affect the conclusions. After accreting 1M⊙ onto a
from applyingthemodel toobserved neutronstars are pre- 1.4M⊙ neutron star we have Mg = 2.05M⊙, which exceeds
sented in Section 3; some further discussion of the impli- the maximum mass for a significant number of equations
cations (Section 4) and an outline of the main conclusions of state. To be conservative, I shall just assume that accre-
(Section 5) follow. tion does not lead to thecollapse of theneutron starinto a
(cid:13)c 0000RAS
2 Ralph Wijers
black hole rather than use the maximum mass as an extra 2×10−8M⊙yr−1,theEddingtonrateforelectronscattering
constraint. opacity on material with hydrogen mass fraction X = 0.7
onto a 10km neutron star. Pulsars with initially high fields
retainnomemoryoftheirinitialspinperiodsoncetheyreach
2.2 Magnetic field evolution fields below about 1010G when they are evolved using the
above equations. Since the only pulsars to which spin con-
Sinceboth themagnetic fields and theamounts of accreted
straintsareappliedherehavelowerfieldsthanthis,theini-
massofneutronstarsvarybymanypowersoften,anymodel
tial period does not matter, and all pulsars are started at
relating thetwo will almost necessarily bea power law:
P = Peq. For very short-lived accretion phases that start
B ∆M −β when the pulsar is still spinning very rapidly (e.g. Brown
= 1+ . (4)
B0 (cid:16) Mc (cid:17) 1995) this is not correct, because the pulsar will then eject
B and B0 are the current and initial field, ∆M is the ac- mass thrown towards it and never accrete. This more com-
creted (baryon) mass, and Mc is the characteristic mass at plicated situation is treated elsewhere (Wijers, Braun, &
which field decrease becomes significant. For β = 1 this is Brown 1997).
the well-known scaling relation proposed by Taam & Van Theinitialmagneticfieldisthatofyoungradiopulsars,
denHeuvel(1986)andShibazakietal.(1989).Ihaveadded which at birth have logB0 = 12.3 typically, with a spread
the exponent because the generalisation probably encom- around that of a factor 2. To investigate the influence of
passes effectively all models in which B depends on ∆M theinitialfieldIcomputeallresultsforlogB0 =11.8, 12.3,
only. Once the amount of accreted mass becomes enough and 12.8. This range nominally covers 87% of all pulsars
to affect theradius of the neutron star, one should perhaps in the best-fit initial magnetic field distribution found by
consider changes in the surface field due to conservation of Hartmanetal.(1996).Inassemblingthefinalconstraintson
magnetic flux BR2, but hereI useEq. 4 throughout. theparametersMcandβIusewhichevervalueinthisrange
is least constraining to the accretion-induced decay model.
Therefore all the derived bounds are necessarily somewhat
2.3 Spin evolution statistical in natureratherthanabsolute, because some 7%
of pulsars will lie outside an indicated range on either side.
Another constraint on the decay model is imposed by the
fact that some recycled pulsars are spinning very rapidly,
near their equilibrium period. For low fields and/or low ac- 2.5 Example evolution tracks
cretion rates the accreted mass cannot carry the required
Fig.1showsthelowerleftcorneroftheradiopulsarperiod-
angular momentum to the neutron star. To follow the evo-
magnetic field diagram with the known pulsars from the
lutioninthisphaseIusethetorquemodelfordiscaccretion
Princeton catalogue (Taylor, Manchester, & Lyne 1993, as
by Ghosh & Lamb (1979). Thespin-up rate is given by
updated by Taylor et al. 1995) indicated as open symbols.
P˙ =−3.6×10−4B122/7R66/7I4−51M13./47M˙−6/87P2n(ωs) syr−1,(5) The filled symbols with names are the pulsars that are dis-
cussed below. Their properties are listed in the top part of
where B12 = B/1012G and M˙−8 = M˙/10−8M⊙yr−1. ωs Table 1. For pulsars close to the Hubble line (lower dotted
(called fastness) is the ratio of neutron star spin frequency
curve)thevalueofthemagneticfieldderivedfromtheperiod
to the Kepler frequency of an orbit just outside the mag-
derivative(Bm inTable1andfilledsquaresinFig.1)should
netopause and n(ωs) is the dimensionless torque function be corrected for the proper motion of the pulsar which
which is given approximately by
contributes to the measured period derivative (Shklovskii
n(ωs)=1.391−ωs[4.03(11−−ωωs)s0.173−0.878]. (6) t1via9ml7u0ae;tseCs(Babmeasiilenod,TToanhbolearns1etaats,nsdu&mfiKelldeudl7k5tarrkinamings1l−9e19s4it)nr.aFTnishgve.er1cs)oerarvreeecltoeecsd--
The fastness is related to thespin period by
ity,exceptforPSRJ2317+1439, whereatransversevelocity
ωs =0.43B162/7R618/7M1−.45/7M˙−−83/7P−1. (7) of 70kms−1 has been measured (Camilo, Nice, & Taylor
1996a).
(M1.4 =Mg/1.4M⊙)Sincethephysicsoftheinteractionbe- AllevolutiontracksshowninFig.1areforlogB0 =12.3
tweenaccretiondiscandmagnetosphereisuncertainandthe and M˙−8 =2. The dashed-dotted curves are for β =1 and
expressionsalsodependonthepoorlyunderstoodstructure µc ≡log(Mc/M⊙)=−5 (lower) and µc =−3 (upper). The
oftheaccretion discitself,thereisampleuncertaintyinthe
curveforlowµcpassesnicelythroughthemillisecondpulsar
torque.Forthepresentpurposeitisadequate,however.The
range but misses the fastest ones. For high µc the lowest
value of the critical fastness, ωcr, at which the torque van- fieldsarenotreached(thetracksinFig.1allendat∆M =
ishes(0.349forEq.6)isimportant,sotheeffectofchanging 0.8M⊙).Theintermediatecaseµc =−4(notshown)almost
it is examined below. Setting ωs = ωcr in Eq. 7 fixes a re- follows the standard spinup line (heavy solid curve) and is
lation Peq ∝B6/7 called thespinupline orequilibrium spin acceptable.Thecurveforµc =−3illustratestheimportance
period. of the changing neutron star mass: PSRB1821−24 appears
to be above the spinup line, but that is drawn for a fixed
2.4 Initial conditions neutronstarmassof1.4M⊙,asistraditional.Increasingthe
mass shifts the spinup line to the left enough to reach this
In the calculations all neutron stars start with a gravita- pulsar without exceedingM˙−8 =2.
tional mass, Mg, of 1.4M⊙ (so Mb = 1.56M⊙). The ac- Low fields can be reached with µc = −3 by increas-
cretion rate is taken to be constant, with a usual value of ing β. This is illustrated by the dashed (β =1.5) and solid
(cid:13)c 0000RAS,MNRAS000,000–000
Accretion-induced field decay 3
Figure 1. The lower left corner of the B-P diagram of radio
pulsars.Opensquares denote pulsars,filledsymbols withnames Figure 2. Constraints in the (Mc,β) plane imposed by the
arethosespecificallydiscussedinthispaper.Thelowerdottedline lowest-fieldpulsars
indicates a characteristic age, P/2P˙, of 10Gyr, the heavy solid
line is the standard spinup line. The other curves are evolution
tracksofpulsars,explainedinthetext. accreted by the pulsar to be 0.70M⊙ and 0.79M⊙, respec-
tively.Some properties are listed in Table 1.
For PSRJ2229+2643, I use the measured field Bm be-
(β=2)curves.Thedashedcurveisfullyacceptableforany cause the proper motion has not been measured and there-
of the fast, low-field pulsars, whereas the β = 2 case does fore its Be is only based on a typical value (see above). For
not produce the fast ones because the field has decayed to PSRJ2317+1439 the proper motion is measured, so I use
lowvaluesbeforeenoughmasshasbeenaccretedtospinthe Be. The solid line in Fig. 2 is the locus of parameter values
pulsarup.To shift theevolution track furthertotheleft so thatyieldsthefieldofPSRJ2229+2643forits∆M,andthe
thatPSRB1821−24 canbereached,onecaneitherincrease dashed-dottedcurvethesameforPSRJ2317+1439,bothfor
the accretion rate or increase the value of the critical fast- logB0 =12.3. They are almost the same, since the slightly
ness.Thelatteroptionisillustratedbythedottedcurve,for lower field of PSRJ2317+1439 and its higher ∆M act in
whichωcr =0.7,twicethestandardvalue(seeSection3.2). oppositedirections.UsingBe forPSRJ2229+2643 givesthe
Once accretion stops, pulsars will start spinning down, dottedcurve.Thehighercurveheregivesthestrongestover-
so many pulsars that lie to the right of any track can be all constraint, so I use the solid curve in our final assembly
explained by evolution along that track down to some field ofconstraints,butIstillhavetoaccountforthevariationin
strength and subsequent spindown. For pulsars with ages B0.Thedashedcurvesareasthesolidone,butforlogB0 =
greater than the age of the universe (i.e. below the Hubble 12.8(upper)and11.8(lower).Theyalsoroughlybracketthe
line) this will not work, so they must have emerged from range 0.4–1.5M⊙ for ∆M at fixed logB0 =12.3. Given the
the accretion phase with long periods, close to the current uncertaintyinB0,theentireregionbetweenthetwodashed
values. This poses no problems because for any evolution curves satisfies the requirement that low enough fields be
trackthatpassestotheleftofagivenpulsarthereisonewith attainable.
a lower accretion rate and otherwise the same parameters
thatpassesthroughit.Suchalowaccretionratemayimpose
constraints on binary evolution scenarios, but I shall not 3.2 The fastest pulsars: PSRB1937+21 and
attempt to exploit this here, and use only the requirement B1821−24
that the fastest pulsars beproduced (Section 3.2).
As explained above (Section 2.5), the only spin constraint
applied is that fast enough pulsars can be formed. For
M˙−8 = 2 and ωcr = 0.35 the strongest constraint on the
3 CONSTRAINTS ON β AND MC modelissetbyPSRB1937+21, thelowest-field pulsarclose
to the spinup line. The thick solid line in Fig. 3 indicates
3.1 The weakest fields: PSR J2229+2643 and
the locus of models that pass exactly through its current
J2317+1439
position in the period-field diagram, for logB0 = 12.3. All
The two lowest-field pulsars known are PSR J2229+2643 models below the curve are allowed, those above it are
andJ2317+1439 (Camilo etal.1996a). Bothareinbinaries excluded. The thick dashed lines indicate how the curve
with low-mass white dwarf companions for which Van den changes for logB0 = 12.8 (upper) and 11.8 (lower). The
Heuveland Bitzaraki (1995) estimated the amount of mass next strongest constraints for the same accretion rate are
(cid:13)c 0000RAS,MNRAS000,000–000
4 Ralph Wijers
Table1.Relevantpropertiesofthesystemsused.Radiopulsarsareatthetop,X-raypulsarsatthebottom.Bmisthefield
measuredfromtheradiopulsarspindownrate,Be isthefieldaftercorrectingforpropermotion;forX-raypulsarsitisthe
fieldforwhichthepulsarwouldbeonthespinupline.PulsarcompanionmassesaretakenfromVandenHeuvel&Bitzaraki
(1995). X-raybinaryparametersaretakenfromVanKerkwijk,VanParadijs&Zuiderwijk(1995). M˙−8 isestimated from
theX-rayluminosity.
name Pspin Porb companion logBm logBe M˙−8 τX ∆M
(s) (d) type mass(M⊙) (G) (G) (104yr) (M⊙)
PSRJ0218+4232 0.00232 2.03 HeWD 0.20 8.63
PSRJ1022+1001 0.0165 7.8 COWD >0.45 8.92 8.8 0.01–0.045
PSRB1820−30A 0.00544 single 9.64
PSRB1821−24 0.00305 single 9.35
PSRB1831−00 0.521 1.81 HeWD? 0.20? 10.94 0.80
PSRB1937+21 0.00155 single 8.61
PSRB1957+20 0.00161 0.382 ? >0.02 8.22
PSRJ2145−0750 0.0161 6.84 COWD >0.45 8.84 8.6 0.01–0.045
PSRJ2229+2643 0.00298 93.0 HeWD 0.30 7.88 7.6 0.70
PSRJ2317+1439 0.00345 2.46 HeWD 0.21 7.97 7.8 0.79
SMCX-1 0.717 3.89 B0Ib 15. 12.0 4 2 8×10−4
LMCX-4 13.51 1.41 O7III–V 16. 13.6 5 1 5×10−4
CenX-3 4.84 2.09 O6.5II–III 19. 12.7 1 1 1×10−4
HerX-1 1.24 1.7 A–FIV 2.3 11.7 0.2 30 6×10−4
provided by PSRB1957+20 and J0218+4232. Their curves
forlogB0 =12.3nearlycoincidewiththeupperdashedline
andhavenotbeenplottedseparately.Fornocurveshownis
the required amount of accreted mass greater than 0.8M⊙,
and for the thick solid line it is about 0.3M⊙ everywhere
on it. The constraint is tighter for a lower accretion rate
becauseitputsPSRB1937+21closertotheequilibriumpe-
riod. The thin solid curvediffers from the thickone only in
thatM˙−8 =1,butthedifferenceissmall.Thedottedcurve
illustrateswhathappensifthechangeofmassoftheneutron
starisneglected,butallparametersareasforthethicksolid
curve: it leads to an artificially stronger constraint because
it does not allow thespinupline tomovetoshorter periods
as the neutron star mass increases (see Fig. 1).
NextwemustconsiderwhattodowithPSRB1821−24
andB1820−30A, whichareabovethestandardspinupline.
Boththesepulsarsareinglobularclustersanditispossible
that their location is a consequence of a type of evolution
that is unique to that environment (e.g. a dramatic period
of mass transfer following a collision with another star), so
perhaps we should omit them from the discussion. The lo-
cation of PSRB1820−30A could even be a direct result of
contamination of its period derivative due to acceleration
Figure3.Constraintsinthe(Mc,β)planeimposedbythefastest
in the cluster potential (Biggs et al. 1994). But since I use
pulsars.
standard spinup theory, it is appropriate to at least briefly
ask whether their location above the standard spinup line
indicates a flaw in that theory that is serious enough to re- otheroptionistoincreasethecriticalfastnessto0.7byusing
consider its use. a toy model n(ωs)=1.39(1−ωs/0.7)/(1−ωs). Once again
Increasingthemassoftheneutronstarhelpstobringit PSRB1821−24 providesthestrongest constraint, shown by
abovethestandardspinupline,butitturnsoutthatmodels the thin dashed curve (logB0 = 12.3); varying the initial
in which these still relatively high-field neutron stars have field produces similar offsets to the case of PSRB1937+21.
become heavy enough cannot produce the lowest observed Inother words, simple and reasonable adjustmentstostan-
fields. One possible remedy is to accept higher accretion dardspinuptheorycanaccommodatethesetwopulsars,and
rates, as observed in some X-ray pulsars (see Sect. 3.3). If would give stronger constraints. Due to the location of the
we set M˙−8 = 6.2 both pulsars are on the spinup line. It pulsars in globular clusters and the consequent possibility
turns out that the strongest constraint on models is then that they evolved in a manner radically different from disc
set by PSRB1821−24, and nearly coincides with the lower pulsars, it is better not to use those constraints.
dashed line, so it would provide a stronger constraint. An- The weakest constraint from pulsar fastness is the up-
(cid:13)c 0000RAS,MNRAS000,000–000
Accretion-induced field decay 5
per dashed curve or the thin dashed curve displaced up- 1979).Inthismodel,thedonorstarisstillburninghydrogen
wards as appropriate for logB0 = 12.8. Which one is used initscore,butextendedinsizeduetomassloss.Itexpands
matters little once it is combined with theconstraint set in on a nuclear time scale and the mass transfer rate grows
the previous section. The area that satisfies the most con- exponentially with an e-folding time, τX, equal to the time
servativeconstraintsfromFigs.2and3isshowninFig.4as it takesthestar toexpandbyonescale height.Formassive
the hatched region. It is mostly determined by the field of donorsthistimescaleisabout104yr,inagreementwiththe
PSRJ2229+2643, with a minor contribution from the spin estimated X-ray life time from population statistics (Meurs
of PSRB1937+21 at low logMc. & Van den Heuvel 1989). HerX-1 has a subgiant donor of
muchlowermass,soitslifetimeisratherlonger.Thevalues
shown in Table 1 are taken from Savonije. Since the accre-
3.3 X-ray pulsars
tion rate grows exponentially,∆M is well approximated by
The classical high-mass X-ray binaries contain high-field τX times thecurrent accretion rate.
neutron stars. To estimate their amounts of accreted mass The magnetic fields of X-ray pulsars are high, but pre-
we need the accretion rate, which can be derived from the cise values are hard to specify. The estimate in Table 1 is
observedX-rayluminosity,andtheX-raylifetime,forwhich based on the assumption that the pulsar is on the equilib-
modelsarerequired.Thefinallyadoptedvaluesforthefour rium spinup line, but only for HerX-1 is the spinup time
X-ray binaries SMCX-1, LMCX-4, CenX-3, and HerX-1 scale sufficientlyshorterthan τX tojustify thisassumption.
are listed in Table 1. The cyclotron line feature seen in the spectrum of HerX-1
TheX-rayluminositiesweretakenfrom White,Swank, indicatesasurfacefieldof3×1012G(Tru¨mperetal.1978),
& Holt (1983) and are somewhat higher than often quoted. in good agreement with the spinup estimate given that the
This is because X-ray pulsars have hard spectra and usual cyclotron line probes the field very near the neutron star,
fluxesarecitedintherange2–10keV.ButWhiteetal.quote whereas the spinup estimate probes the large-scale dipole
luminosities from 0.5–60keV, which much better represent field.Nocyclotronlineshavebeenseenfromtheotherthree
the total luminosity, as can be seen from the spectra they X-ray pulsars. Since they had lower accretion rates and
show. All four sources are disc accretors and are not ex- thereforelongerequilibriumperiodsinthepastandarespin-
pected to vary too much in flux. For example, a low flux is ningup,theirequilibrium periodsare probablyshorter and
sometimesseenforSMCX-1(e.g.Seward&Mitchell1981), therefore the field strengths are overestimates.
but a 7-year monitoring using the Vela 5B satellite (Whit- If we assume that the four pulsars have had a factor 3
lock&Lockner1994)showsanaverage3–12keVluminosity field decrease each one defines a curve in the (Mc,β) plane
of2×1038ergs−1despitetheoccasionallowstate,quitecon- that is shown in Fig. 4. (A small systematic difference be-
sistent with the 2.5 times higher value for thewider energy tweentheirfieldsandthoseofstandardradiopulsarscannot
range quoted by White et al. The X-ray luminosities were be ruled out, but a factor 10 is too much. A factor 3 is a
convertedtoaccretion ratesassumingthatLX =GMM˙/R, reasonable estimate of what might still be allowed.) The
with M =1.4M⊙ and R=10km. area below and to the right of the curves corresponds to
One might worry that beaming of the X-ray pulsar ra- lower amounts of field decrease and is thus allowed. The
diation causes an overestimate the luminosity because we area above them is excluded. The three curves defined by
are hit by the beam but other positions in the sky are not, LMCX-4, HerX-1, and SMCX-1 are quite similar. I shall
and ourestimates assume isotropic emission. Forradio pul- take the middle one (HerX-1) to fairly represent the X-ray
sars, which haveverynarrow beams, thisclearly a concern. pulsarconstraint.Theallowedregionthusleftnolongercon-
But X-ray pulsar profiles are very different: inspection of tainsanycombinationwithβ =1,sothestandardmodelin
the pulse profiles in White et al. (1983) reveals that signif- whichfielddecayisproportionaltoaccretedmassisalready
icant emission is present over the full 360◦ of pulse phase, ruled out.
and the rms. variation of the intensity over phase is 0% for
CenX-4,5–20% for SMCX-1,20–50% for HerX-1and 50% 3.4 PSRJ2145−0750
for CenX-3. (Ranges are given if the modulation depth de-
pendson energy band.)Sincetheseare verybrightsources, Arecentlydiscoveredclassofmillisecondpulsarswithhigh-
they would be detected even if the phase-average flux were massCO whitedwarf companionsprovidesuswith aprobe
as low as the lowest flux that occurs at any phase. This is of the regime of ∆M in between the X-ray pulsars and the
unlike radio pulsars, many of which go undetected because most strongly recycled pulsars. Van den Heuvel (1994) dis-
their beam is neverpointed towards us. Sincetheflux vari- cusses their evolution in detail and concludes that the pro-
ation in the direction perpendicular to the beam sweep is genitor of the white dwarf was a 1–6M⊙ star that came
unlikely to be very different from the one along the pulse, into Roche contact on the asymptotic giant branch. The
X-ray pulsars are detectable from any direction in the sky. amount of accreted mass is mainly from wind accretion
This means that (i) the phase-averaged flux we see is un- and increases from 0.01 to 0.045M⊙ with increasing donor
likelytodeviatemuchfromthemeanovertheentiresphere mass.Forthehighest,leastconstrainingamountofaccreted
aroundthepulsarand(ii)thedeviationcouldbepositiveor mass,andforlogBm =8.84 forPSRJ2145−0750 (basedon
negative, i.e. in a sample of pulsars we will not systemati- P˙ = 2.98×10−20, F. Camilo, private communication) the
cally overestimate the luminosity. Therefore, the estimated curve of acceptable parameter pairs is shown in Fig.4 for
amounts of transferred mass are not systematically biased logB0 = 12.3 (solid), 12.8 (upper dotted), and 11.8 (lower
upwardsdue tobeaming. dotted). The lower curve just grazes the still acceptable re-
To estimate the X-ray life times I use the model of gion at (−3.2,1.6), so if one wants to keep a field decay
beginning atmospheric Roche lobe overflow (Savonije 1978, model from this class these are its parameters. More likely,
(cid:13)c 0000RAS,MNRAS000,000–000
6 Ralph Wijers
(1995) claim that its position at high field is good evidence
for accretion-induced collapse of a white dwarf. In this sce-
nario, the accreting object in the binary was a white dwarf
while most of the donor mass was accreted, but near the
endofaccretionitreachedtheChandrasekharmassandcol-
lapsed to a neutron star, which then accreted little enough
tokeepahighfield.Forthestandardβ =1modeltheyused
thismaystillbereasonable,sincetheaccretedamountafter
collapse could be 0.1M⊙. This is a fair fraction of the total
transferred (∼ 0.8M⊙) so the moment of collapse need not
be tuned too delicately. But the most tolerable model still
left now (β=1.6,µc =−3.2) limits theamount of accreted
mass to0.01M⊙,even ifwe allow thepulsar tostart with a
1013Gfield.Thisdoesrequirethecollapsetoberatherfinely
timedveryneartheendofaccretion,andmakesthescenario
quite unattractive. Even if one were willing to accept this
fine tuning for PSRB1831−00, there is still the following
coincidence: why should it be the only short-period binary
pulsarwithsuchahighfield?Afterall,theremusthavebeen
less fortunately timed cases like it, in which the collapse
occurred less close to the end of mass transfer. The result-
ing neutron stars would have lower fields, between that of
PSRB1831−00 and the canonical low values, and therefore
Figure 4.Summaryofallconstraints inthe(Mc,β)plane.The
live longer and be more abundant. Note that this problem
blackdotmarksthelocationofthemodelthatisstillmarginally
consistentwithallconstraintsexcept PSRB1831−00. occurs no matter which parameters one chooses for the de-
cay model. Consequently, I feel that one should accept the
factthatithasaccretedsignificant amountsofmass,ashas
onecannotinreasonstretchalltheconstraintstotheirlim- 4U1626−67. This is of course fatal to the decay model, as
its simultaneously and the model is inconsistent with the illustrated by the lowest curves in Fig. 4, which define the
whole body of available data. The best prospect for ruling regionofacceptableparameterstofitPSRB1831−00(using
it out altogether comes from PSRJ2145−0750 and its twin ∆M =0.8M⊙ and logB0 =11.8,12.3,12.8).
sister, PSRJ1022+1001 (Camilo et al. 1996a). Both have AIC has at various times in the past been called on to
significantestimated proper-motioncorrectionstothemag- rescue the then popular field decay model, by providing an
netic field (see Fig. 1 and Table 1), so the lower values of alternativeformationmechanismforthefewobjectsthatob-
the corrected fields will push the curve up and remove it viouslyfailedtofitthemodel.SoitwaswhenHerX-1failed
further from the area allowed by the constraints. Also, not to fit the rapid exponential decay of neutron star magnetic
all will have had the maximum allowed accreted mass. To fields, but Verbunt et al. (1990) showed that AIC cannot
indicatethestrongeffectofthisIshowtheacceptablecurve provide a viable alternative in that case and that therefore
for PSRJ2145−0750 for the same parameters as the solid HerX-1doesdisprovethemodel. Itwill remain amatterof
curveexceptthat∆M =0.02M⊙ (dashed-dotted).Itleaves taste in each case whether to view the lone strange object
no acceptable parameter combinations. as an exception that proves the model or a nail its coffin.
But in the case of PSRB1831−00 and 4U1626−67 the ex-
ceptionsare onceagain nails, if not theonly ones, andAIC
4 DISCUSSION cannotchangethis(Thisargumentwasalreadypartlygiven
byVerbuntet al. 1990.)
4.1 PSRB1831−00 and 4U1626−67:
Accretion-induced collapse?
4.2 Constraints on field only
In the previous section I have deliberately not included
two objects that have been adduced as good evidence Since the spin period constraints did not contribute too
againstfielddecayproportionaltoaccretedmassinthepast: muchtoconstraining thefielddecaymodel, itispossibleto
PSRB1831−00 andtheX-raypulsar4U1626−67.Theages capture most of the argument in a straight plot of amount
andorbitalparametersofbothsuggestthattheneutronstar offieldchangeversusaccreted mass(Fig.5).Each diamond
hasaccretedatleast0.5M⊙,yetthefieldsarehigh(Verbunt represents one object whose name is plotted alongside it.
et al. 1990). The reason is that at various times accretion- Thesizeofthediamondrepresentstheerrorinthelocation
induced collapse (AIC) of a white dwarf has been proposed of the object. Most of the ‘errors’ in the quantities ulti-
as a method of making these, allowing the neutron star to matelyincludeuncertaintiesinthemodelsusedtointerpret
haveaccreted less mass than it seems, so I wanted to make the data from which the quantities are derived and they
the case against the decay model without them. Also, both therefore are indicative of the author’s personal assessment
havepeculiarlylowcompanionmassessotheymayreallybe of those models rather than well-defined statistical quanti-
different. ties. The error in the ratio B/B0 was taken to be 0.5 dex
But let us briefly revisit PSRB1831−00. It has B = andmainlyrepresentsthevariationofinitialmagneticfields
8.7×1010GandP =1.8d.VandenHeuvelandBitzaraki of neutron stars; this range nominally covers 87% of initial
orb
(cid:13)c 0000RAS,MNRAS000,000–000
Accretion-induced field decay 7
advancing accretion-induced collapse for the formation of
PSRB1831−00 (sect.4.1) are even greater for 4U1626−67,
where the amount of accreted mass needs to be reduced by
3ordersofmagnitudefrom thetotalamountofmasstrans-
fer that has occurred in thesystem before it comes close to
fittinganyofthemodels.Butmasstransferinthissystemis
stillongoingandverylong-lived,sothisneutronstarwould
have to have formed almost literally yesterday by AIC to
savethe accretion-decay model.
Itistemptingtoconcludefrom Fig.5thatasignificant
amount ofaccreted mass, while it does not guaranteemuch
field decay, is at least a necessary condition for it, because
there appear to be no objects in the lower left corner. But
that conclusion is not valid because this absence may well
be due to poor chances of detecting such sources. Imagine
a neutron star with very little accreted mass that has had
much field decay. If it were not accreting it could only be
visible as a radio pulsar. But for it to be above the death
lineitwouldhavetohavebeenspunup,whichisnotpossi-
ble without significant mass accretion. If it were an accret-
ingsource,chancesofdetectionaresmallagain becausethe
small amount of accreted mass implies that either the ac-
Figure5.Theaccretedmassversusmagneticfieldforallobjects
cretion phasehas tobe veryshort or theaccretion rate has
in Table 1 except those that constrain the model through their
to bevery low.
spinperiod.Thedashedcurvesaredecaymodelswithβ=1and
µc=−4(top)and−5(bottom).Thesolidcurveisforthemodel
markedwithablackdotinFig.4(β=−1.6,µc=−3.2).
4.3 Further implications for models
ThedetailedmodelbyRomani(1993) doespredictfieldde-
fields (Sect.2.4). The amounts of accreted mass for the X- cayproportionaltoaccretedmassdowntoabottomvalueof
ray pulsars are derived partly from theory and partly from around108GwithMc =10−5−10−4M⊙,aslongastheac-
data.Thoseofthemillisecondpulsarsarerelatedviatheory cretionrateishigh.Sincealltheobjectsusedinthepresent
to system parameters that are accurately known observa- paper have accreted most of their mass at rates not far be-
tionally, so theerror is mostly uncertainty in thetheory.In low the Eddington rate this model as it stands is excluded
all cases, I have assigned an error of a factor 2 either way bythe data.
to the estimated ∆M, which should be fairly generous. For The boundaries I have drawn are statistical in na-
PSRJ1022+1001 and J2145−0750, the theoretical estimate ture due to the possible variations in initial magnetic field
itself already spans a factor 2 each way from the mean of (Sect.2.4). But the range used everywhere covers almost
0.02M⊙(Table1),soIhaveusedatotalerrorofafactor4ei- 90% of initial fields, and for every type of constraint there
therway for their ∆M. Given thegenerous errors used one are at least 2 objects which providea nearly equally strong
should really demand that any model curve pass through versionofthatconstraint.Thereforeitishighlyunlikelythat
80% of all the diamonds. For any diamond that is widely we have condemned the model unjustly because all pulsars
missed,onewillhavetofindanalternativemodelthatputs used to constrain it are untypical of the average in a direc-
theobject in a completely different position. tion that is unluckyfor themodel.
Thefirst strikingimpression from Fig.5 isthatthefull It is of course possible to save the model by varying
rangeofaccretedmassesisrepresentedbyobjectswithlittle the parameters β and Mc between pulsars, but this would
ornofielddecay,andthefullrangeofamountsoffielddecay largely remove its attraction unless one had a prescription
isrepresentedbyobjectsthathaveaccreted around0.5M⊙. forwhatparameterstogivetowhichkindofpulsar.Doingso
This bodes ill for any model that demands a strict relation would merely reinforce the main conclusion of the present
between the two quantities. A few models are indicated as paper, namely that other parameters than accreted mass
well:thedashedcurvesrepresentthestandardversionofthe alone influence significantly the decay of a neutron star’s
model, β =1. They lie either too low for the X-ray pulsars magnetic field.
ortoohighforPSRJ1022+1001andJ2145−0750.Thesolid
curverepresentsthemarginal model indicated bythe black
dotinFig.4.Itjustgrazesthecornersofmanyerrorboxes,
5 CONCLUSION
and is already in trouble at the right of the diagram, since
its predicted fields there are almost too low for the lowest- I have explored the consequences of a popular neutron star
field pulsars known, but many pulsars exist at that same magnetic-fielddecaymodelinwhichthedecayisdetermined
amount of accreted mass and ten times higher fields (like solelybytheamountofaccretedmass.Theresultisthatno
PSRJ0218+4232). model of this type (Eq. 4) can satisfy all the known con-
The really difficult objects for the accretion-decay straints derived from the fields, estimated amounts of ac-
model are 4U1626−67 and PSRB1831−00. They cannot creted mass, and spin periods of a variety of recycled radio
possibly satisfy any model of this type. The difficulties of pulsars and X-ray binaries. Only if the constraints are all
(cid:13)c 0000RAS,MNRAS000,000–000
8 Ralph Wijers
stretched to their limit simultaneously and PSRB1831−00 Seward, F. D. & Mitchell, M. 1981, ApJ243, 736
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